1. Continuous Probability Distributions  Continuous Random Variable  A random variable whose space (set of possible values) is an entire interval of numbers  Probability Density Function (or pdf)  The pdf, denoted f( x ), describes the distribution of probability across the set of possible values.  The probability that the random variable X takes on a value between a and b is equal to the area under f( x ) between a and b.

2. Probability Density Function  The pdf of a continuous random variable X has the following properties: 1. f( x ) ≥ 0, for every x  S 2. - ∞  ∞ f( x ) dx = 1 3. P( x  A ) = A  f( x ) dx

3. Probability Density Function  So we have that:  P( a < x < b ) = a  b f( x ) dx  P( X = a ) = a  a f( x ) dx = 0  P( a ≤ x ≤ b ) = P( a < x < b )

4. Histogram  A “connected” bar plot with bar height proportional to the frequency of the associated class.  Can be very useful for estimating a pdf.  Discrete Data  Each distinct outcome is marked on horizontal axis.  Bar is plotted atop each distinct outcome with bar height equal to frequency or relative frequency or that outcome.

5. Histogram  Continuous Data  Construct a frequency table  Partition data into classes and tabulate the frequency for each class.  Use frequency table to plot histogram  Mark class boundaries on horizontal axis and plot a bar on top of each class with height equal to the frequency or relative frequency of data in that class.

6. Cumulative Distribution Function  Called the cdf or distribution function  The cdf of a continuous random variable X is given by: F( x ) = P( X ≤ x ) = - ∞  x f( t ) dt  Consider that  P( a < X < b) = P( a ≤ x ≤ b ) = F( b ) – F( a )  P( X > a ) = 1 – P( X ≤ a ) = 1 – F( a )

7. Cumulative Distribution Function  Using the fundamental theorem of calculus, it can be shown that:

8. Percentile  Let p be a number between 0 and 1, and let X be a continuous random variable with pdf f( x ) and cdf f( x ).  The 100·p th percentile is the number such that F(  p )=p.  Solve the following equation for  p :

9. Mathematical Expectation  The mathematical expectation of a continuous random variable X with pdf f( x ) is:  = E[ X ] = - ∞  ∞ x f( x ) dx  E[ X ] is also called the mean or expected value of X

10. Variance  The variance of a continuous random variable X with pdf f( x ) is:   = E[ ( X –  )  ] = - ∞  ∞ ( x –  )  f( x ) dx = - ∞  ∞ ( x –  )  f( x ) dx = E[ X  ] – (E[ X ])  = E[ X  ] – (E[ X ])   measure of spread in f( x )

11. Expected Value of a Function  The expected value of the function h( x ) of a continuous random variable X with pdf f( x ) is:  = E[ h( X ) ] = - ∞  ∞ h( x ) f( x ) dx  Note that E[ h( X ) ] might not exist.

12. Continuous Uniform Distribution  Uniformly distributes the probability across the sample space.  If X is a continuous random variable on the interval [a,b], then  The pdf of X is: f( x ) = 1 / ( b – a ), for a ≤ x ≤ b

13. Continuous Uniform Distribution  If X is a continuous random variable on the interval [a,b], then  The cdf of X is: F( x ) = ( x – a ) / ( b – a ), for a ≤ x ≤ b  The mean and variance of X are:  = E[ X ] = ( a + b ) / 2   = ( b – a )  / 12

14. Exponential Distribution  Can be used to describe the waiting time between successive events in a Poisson process with mean  Can be used to describe the waiting time between successive events in a Poisson process with mean  If X is an exponential random variable from a Poisson process with mean, then  The pdf of X is: f( x ) = e - x , for x ≥ 0

15. Exponential Distribution  If X is an exponential random variable from a Poisson process with mean, then  The cdf of X is: F( x ) = P ( X ≤ x ) = 1 - e - x, for x ≥ 0  So P ( X > x ) = 1 – F ( x ) = e - x  The mean and variance of X are:  = 1/  = 1/ 

16. Exponential Distribution  Memoryless Property  P( X > k + j | X > k ) = P( X > j )  Percentiles of an Exponential  The p th percentile of an exponential random variable with mean 1/ is:  p = -1/ ln( 1 – p )

17. Normal Distribution  Commonly occurring distribution in nature and experimental settings.  symmetric, bell-shaped  A normal random variable X with mean  and variance  2  is denoted: X ~ N( ,  2 )  has pdf:

18. Standard Normal Distribution  A standard normal random variable is denoted Z and has distribution Z ~ N( 0, 1 ).  The pdf and cdf of a standard normal random variable are denoted  ( z ) and  ( z ) respectively.  Table A.3 contains probability values associated with  ( z ).  Note that  ( z ) = 1 -  ( -z )

19. Standard Normal Distribution  z  Notation  The value of Z that has  probability to its right is denoted z , so P( Z > z  ) = . by symmetry, P( Z z  ) =   Percentile  p = z  p

20. Non-Standard Normal Distribution  Any normal random variable X ~ N( ,  2 ) can be “standardized” into a Z by: Z = ( x –  ) /   Percentile  p =  + z  p 

21. Normal Approximation of Discrete Distributions  Many discrete random variables can be approximated by the normal distribution  A continuity correction of ±½ is required when estimating a discrete probability with the normal distribution

22. Normal Approximation of the Binomial Distribution  Let X be a binomial random variable with sample size n and probability of success p. If the sample size is sufficient ( np ≥ 5 and nq ≥ 5 ), then X can be approximated by a normal distribution with the same mean  =np and variance  2 =npq. X ~ B( n, p )  N( np, npq ) PDF for Graphs: Binomial n=15Binomial n=15, Binomial n=50, Binomial p=0.25 Binomial n=50Binomial p=0.25 Binomial n=15Binomial n=50Binomial p=0.25

26. Normal Approximation of the Poisson Distribution  Let X be a Poisson random variable with sufficiently large mean, then X can be approximated by a normal distribution with the same mean  =  2 =npq. X ~ P( )  N(, ) PDF for Graphs:Poisson (  = 1, 5, 10, 15 ) Poisson (  = 1, 5, 10, 15 )Poisson (  = 1, 5, 10, 15 )

28. Empirical Rule For data that is approximately normal in distribution (bell-shaped),  68% of data values fall within 1 standard deviation of the mean,  95.4% of data values fall within 2 standard deviation of the mean,  99.7% of data values fall within 3 standard deviation of the mean,

29. x - 3s x - 2s x - s x x + 2s x + 3s x + sx + s 68% within 1 standard deviation 34% 95% within 2 standard deviations 99.7% of data are within 3 standard deviations of the mean 0.1% 2.4% 13.5% The Empirical Rule (applies to bell-shaped distributions )

30. Identifying Unusual Observations  Range Rule of Thumb:  Empirical rule says 95% of observations should fall within 2  of the mean. Observations outside of  ± 2  are considered unusual.  Probability Approach:  The probability of the observed outcome or more extreme can be useful for identifying unusual observations. For example, if X is the observed outcome:  X is unusually high if P(x or more) is less than 0.05  X is unusually low if P(x or less) is more than 0.05

31. Probability Plots  A probability plot can be useful for comparing the distribution of one sample data set to another.  If both data sets have the same sample size, then plot the order statistics from each sample against each other in a scatter plot.  Otherwise, plot order statistics from the smaller sample against corresponding sample percentiles from the other sample.  Note: The i th smallest observation is taken to be the (100*(i-½)/n) th sample percentile.

32. Measures of Relative Position  Percentile  The k th percentile (P k ) separates the bottom k% of data from the top (100-k)% of data.  The location of P k in the order statistics is:

33. Interpretation of Probability Plots  Probability plot for comparing 2 sample data sets:  A straight line with slope 1 and y-intercept 0 indicates identical sample distributions.  A slope greater than 1 indicates that x is less variable than y.  A slope less than 1 indicates that x is more variable than y.  A y-intercept different from 0 indicates that the two samples have a different mean.

34. Probability Plots  A probability plot can be useful for comparing the distribution of sample data to a specified probability distribution.  Order statistics (or sample percentiles) of the sample are plotted against the corresponding percentiles of the probability distribution of interest.  Called the “Normal Probability Plot” when sample data is compared to normal percentiles.

35. Simulating Data from a Continuous Probability Distribution  Theorem  Let Y be U(0,1), a continuous uniform random variable on the interval (0,1). Let F( x ) have the properties of a cdf, then X = F -1 ( Y ) is a continuous random variable with cdf F( x ).  So, random data can be generate from any cdf that has an inverse function by generating random U(0,1) data and transforming it into F( x ) data.